Toms Shoes offers new parents at least eight weeks paid time off. It adopted the policy several years ago, but Mycoskie said he noticed that some new dads feared that taking time off would hurt their careers.

Mycoskie said he believes it's a problem that the U.S. is the only developed country that doesn't guarantee paid time off for new parents.

"So much of our strength comes from our families, and if we're not investing in our families, we're not going to be strong in business or in culture," Mycoskie said.

For companies, he added, benefits like paternity leave can also help attract talent, particularly millennials. Appealing to the next generation is "crucial" to the future of big businesses, and those workers will be looking for perks like family leave.

Federal law guarantees new parents 12 weeks of time off, but does not require employers to offer pay.